The adsorption of charge rigid macromolecules, such as proteins from solution, on mixed (charged and neutral) lipid membranes is affected by several important factors. First, the mobile lipids in the membrane may rearrange, and demix locally to match the charge density of the apposed macromolecule, thus lowering the adsorption free energy. On the other hand, the (electrostatic) interaction between adsorbed macromolecules tends to lower the saturation coverage of the membrane. Additional factors, such as non-ideal lipid demixing or an elastic membrane response, enhanced by the presence of the charged macromolecules, may be at the base of the experimentally observed formation of high density protein domains and lateral macro-phase separation in lipid membranes. (C) 2002 Elsevier Science B.V. All rights reserved.
Macroion adsorption on a mixed, fluid, lipid membrane containing oppositely charged lipids induces local changes in lipid composition at the interaction zones, and gradients at their boundaries. Including these effects in the free energy of the macroion-dressed membrane we derive its spinodal equation, and show that nonideal lipid mixing can lead to (lipid-mediated) attraction between macroions and lateral phase separation in the composite membrane. The critical nonideality for this transition is substantially smaller than that of the bare lipid membrane, decreasing with macroion size and charge. That is, the lipid membrane is destabilized by macroion adsorption.
We calculate the forces required to package (or, equivalently, acting to eject) DNA into (from) a bacteriophage capsid, as a function of the loaded (ejected) length, under conditions for which the DNA is either self-repelling or self-attracting. Through computer simulation and analytical theory, we find the loading force to increase more than 10-fold (to tens of piconewtons) during the final third of the loading process; correspondingly, the internal pressure drops 10-fold to a few atmospheres (matching the osmotic pressure in the cell) upon ejection of just a small fraction of the phage genome. We also determine an evolution of the arrangement of packaged DNA from toroidal to spool-like structures.
We present a molecular-level theory for amphiphile packing in linear micelles, focusing on the early stages of micellar elongation, i.e., on small and ``intermediate-size'' micelles, whose endcaps are not yet molded into a final shape. The internal free energy of a micelle of given size and shape is expressed as an integral over local molecular packing free energies in different regions of the micelle. The free energy per molecule is expressed as a sum of interfacial (''opposing forces'') and chain conformational contributions, both depending on the local geometry. The equilibrium shape and energy of the micelle is determined by functional minimization of the total free energy. For amphiphiles exhibiting strong preference for packing in the cylindrical geometry, we show that the early stages of growth involve an energetic barrier, resulting in a ``gap'' in the micellar size distribution. That is, at low total amphiphile concentrations only small (globular) micelles appear in solution. Their concentration reaches a well-defined saturation value, beyond which, all added amphiphiles are incorporated in long micelles, whose ``non-interacting'' endcaps are well separated by the cylindrical middle part. This, ``second CMC'' behavior is demonstrated by numerical calculations of micellar size distributions and average aggregation numbers as a function of the total concentration. The conditions necessary for the appearance of a second CMC are analyzed theoretically, with explicit reference to the underlying molecular packing characteristics. In particular, it is shown that a necessary condition for the appearance of a sharply defined second CMC is that the endcap energies (of at least some) of the small or intermediate-size micelles must be considerably lower than the asymptotic (long micelle) value of this quantity. The diameter of the minimal, spherical micelles, as well as that of the final endcaps, is found to be larger than the diameter of the cylindrical body of the very long micelles. Our results are in good qualitative agreement with recent cryo-TEM imaging studies of micellar shape and growth, as well as with previous (less direct) experiments revealing second CMC behavior.
The association of two species to form a bound complex, e.g., the binding of a ligand to a protein or the adsorption of a peptide on a lipid membrane, involves an entropy loss, reflecting the conversion of free translational and rotational degrees of freedom into bound motions. Previous theoretical estimates of the standard entropy change in bimolecular binding processes, Delta S degrees, have been derived from the root-mean-square fluctuations in protein crystals, suggesting Delta S degrees approximate to -50 e.u., i.e., T Delta S degrees approximate to -25 kT = -15 kcal/mol. In this work we focus on adsorption, rather than binding processes. We first present a simple statistical-thermodynamic scheme for calculating the adsorption entropy, including its resolution into translational and rotational contributions, using the known distance-orientation dependent binding (adsorption) potential. We then utilize this scheme to calculate the free energy of interaction and entropy of pentalysine adsorption onto a lipid membrane. obtaining T Delta S degrees approximate to -1.7 kT approximate to -1.3 kcal/mol. Most of this entropy change is due to the conversion of one free translation into a bound motion, the rest arising from the confinement of two rotational degrees of freedom. The smaller entropy loss in adsorption compared to binding processes arises partly because a smaller number of degrees of freedom become restricted, but mainly due to the fact that the binding potential is much ``softer.''
The cooperative condensation of DNA and cationic liposomes to form ordered aggregates in aqueous solution is associated with the release of partially bound counterions. We directly determine the extent of counterion release by separating the supernatant from the precipitated condensates, measuring the conductivity of the solution before and after the phase transition. The extent of counterion release is calculated for a range of lipid/DNA concentration ratios based on the nonlinear Poisson-Boltzmann theory. Both experiment and theory show maximal, essentially complete, release of counterions at the isoelectric point, where the positive (lipid)/negative (DNA) charge ratio is 1:1. Furthermore, at this point the entropic contribution to the condensation free energy is maximal and dominant.
The adsorption free energy of charged proteins on mixed membranes, containing varying amounts of (oppositely) charged lipids, is calculated based on a mean-field free energy expression that accounts explicitly for the ability of the lipids to demix locally, and for lateral interactions between the adsorbed proteins. Minimization of this free energy functional yields the familiar nonlinear Poisson-Boltzmann equation and the boundary condition at the membrane surface that allows for lipid charge rearrangement. These two self-consistent equations are solved simultaneously. The proteins are modeled as uniformly charged spheres and the (bare) membrane as an ideal two-dimensional binary mixture of charged and neutral lipids. Substantial variations in the lipid charge density profiles are found when highly charged proteins adsorb on weakly charged membranes; the lipids, at a certain demixing entropy penalty, adjust their concentration in the vicinity of the adsorbed protein to achieve optimal charge matching. Lateral repulsive interactions between the adsorbed proteins affect the lipid modulation profile and, at high densities, result in substantial lowering of the binding energy. Adsorption isotherms demonstrating the importance of lipid mobility and protein-protein interactions are calculated using an adsorption equation with a coverage-dependent binding constant. Typically, at bulk-surface equilibrium (i.e., when the membrane surface is ``saturated'' by adsorbed proteins), the membrane charges are ``overcompensated'' by the protein charges, because only about half of the protein charges (those on the hemispheres facing the membrane) are involved in charge neutralization. Finally, it is argued that the formation of lipid-protein domains may be enhanced by electrostatic adsorption of proteins, but its origin (e.g., elastic deformations associated with lipid demixing) is not purely electrostatic.
The loss of conformational freedom experienced by lipid chains in the vicinity of one, or two, impenetrable walls, representing the surfaces of hydrophobic transmembrane proteins, is calculated using a mean-field molecular-level chain packing theory. The hydrophobic thickness of the protein is set equal to that of the unperturbed lipid membrane (i.e., no ``hydrophobic mismatch''). The probability distributions of chain conformations, at all distances from the walls, are calculated by generating all conformations according to the rotational-isomeric-state model, and subjecting the system free energy to the requirement that the hydrophobic core of the membrane is liquid-like, and hence uniformly packed by chain segments. As long as the two protein surfaces are far apart, their interaction zones do not overlap, each extending over several molecular diameters. When the interaction zones begin to overlap, inter-protein repulsion sets in. At some intermediate distance the interaction turns strongly attractive, resulting from the depletion of (highly constrained) lipid tails from the volume separating the two surfaces. The chains confined between the hydrophobic surfaces are tilted away from the walls. Their tilt angle decreases monotonically with the distance from the walls, and with the distance between the walls. A nonmonotonic variation of the lipid-mediated interaction free energy between hydrophobic surfaces in membranes is also obtained using a simple, analytical, model in which chain conformations are grouped according to their director (end-to-end vector) orientations.
We present a theoretical analysis of the phase behavior of solutions containing DNA, cationic lipids, and nonionic (helper) lipids. Our model allows for five possible structures, treated as incompressible macroscopic phases: two lipid-DNA composite (lipoplex) phases, namely, the lamellar (L-alpha(C)) and hexagonal (H-II(C)) complexes; two binary (cationic/neutral) lipid phases, that is, the bilayer (L-alpha) and inverse-hexagonal (H-II) structures, and uncomplexed DNA. The free energy of the four lipid-containing phases is expressed as a sum of composition-dependent electrostatic, elastic, and mixing terms. The electrostatic free energies of all phases are calculated based on Poisson-Boltzmann theory. The phase diagram of the system is evaluated by minimizing the total free energy of the three-component mixture with respect to all the compositional degrees of freedom. We show that the phase behavior, in particular the preferred lipid-DNA complex geometry, is governed by a subtle interplay between the electrostatic, elastic, and mixing terms, which depend, in turn, on the lipid composition and lipid/DNA ratio. Detailed calculations are presented for three prototypical systems, exhibiting markedly different phase behaviors. The simplest mixture corresponds to a rigid planar membrane as the lipid source, in which case, only lamellar complexes appear in solution. When the membranes are ``soft'' (i.e., low bending modulus) the system exhibits the formation of both lamellar and hexagonal complexes, sometimes coexisting with each other, and with pure lipid or DNA phases. The last system corresponds to a lipid mixture involving helper lipids with strong propensity toward the inverse-hexagonal phase. Here, again, the phase diagram is rather complex, revealing a multitude of phase transitions and coexistences. Lamellar and hexagonal complexes appear, sometimes together, in different regions of the phase diagram.
We present a molecular-level theory for lipid-protein interaction and apply it to the study of lipid-mediated interactions between proteins and the protein-induced transition from the planar bilayer (L-alpha to the inverse-hexagonal (H-II phase. The proteins are treated as rigid, membrane-spanning, hydrophobic inclusions of different size and shape, e.g., ``cylinder-like,'' ``barrel-like,'' or ``vase-like.'' We assume strong hydrophobic coupling between the protein and its neighbor lipids. This means that, if necessary, the flexible lipid chains surrounding the protein will stretch, compress, and/or tilt to bridge the hydrophobic thickness mismatch between the protein and the unperturbed bilayer. The system free energy is expressed as an integral over local molecular contributions. the latter accounting for interheadgroup repulsion, hydrocarbon-water surface energy, and chain stretching-tilting effects. We show that the molecular interaction constants are intimately related to familiar elastic (continuum) characteristics of the membrane, such as the bending rigidity and spontaneous curvature, as well as to the less familiar tilt modulus. The equilibrium configuration of the membrane is determined by minimizing the free energy functional, subject to boundary conditions dictated by the size, shape. and spatial distribution of inclusions. A similar procedure is used to calculate the free energy and structure of peptide-free and peptide-rich hexagonal phases. Two degrees of freedom are involved in the variational minimization procedure: the local length and local tilt angle of the lipid chains. The inclusion of chain tilt is particularly important for studying noncylindrical (for instance, barrel-like) inclusions and analyzing the structure of the H-II lipid phase; e.g., we find that chain tilt relaxation implies strong faceting of the lipid monolayers in the hexagonal phase. Consistent with experiment, we find that only short peptides (large negative mismatch) can induce the L-alpha –> H-II transition. At the transition, a peptide-poor L-alpha phase coexists with a peptide-rich H-II phase.
The phase behavior of a solution containing a mixture of large and small, cross-bridging (''two-sided sticker'') particles is studied using a lattice model analyzed with the aid of mean-field calculations and Monte Carlo simulations. Neither the large nor the small particles interact with each other (except for excluded volume effects). However, the small particles can adsorb onto the surface of one or two large particles, in the latter case providing a cross-bridge, i.e., an adhesive bond, between the large particles. The formulation of the model is motivated by experimental studies involving aqueous solutions of vesicles (the large particles) and biotin-avidin-biotin cross-bridges. This system exhibits a first-order phase transition from a dilute to a condensed phase of vesicles once the average number of stickers per vesicle exceeds a certain threshold value. The statistical thermodynamic description of the system becomes particularly simple upon (Legendre) transformation from the two-component canonical ensemble to a ``mixed'' ensemble involving a constant chemical potential of the cross-bridge particles. The phase separation behavior of the system is calculated for two sets of molecular parametrs, revealing good qualitative agreement with relevant experiments.
We develop a statistical thermodynamic model for the phase evolution of DNA-cationic lipid complexes in aqueous solution, as a function of the ratios of charged to neutral lipid and charged lipid to DNA. The complexes consist of parallel strands of DNA intercalated in the water layers of lamellar stacks of mixed lipid bilayers, as determined by recent synchrotron x-ray measurements Elastic deformations of the DNA and the lipid bilayers are neglected, but DNA-induced spatial inhomogeneities in the bilayer charge densities are included. The relevant nonlinear Poisson-Boltzmann equation is solved numerically, including self-consistent treatment of the boundary conditions at the polarized membrane surfaces. For a wide range of lipid compositions, the phase evolution is characterized by three regions of lipid to DNA charge ratio, rho: 1) for low rho, the complexes coexist with excess DNA, and the DNA-DNA spacing in the complex, d, is constant; 2) for intermediate rho, including the isoelectric point rho = 1, all of the lipid and DNA in solution is incorporated into the complex, whose inter-DNA distance d increases linearly with rho; and 3) for high rho, the complexes coexist with excess liposomes (whose lipid composition is different from that in the complex), and their spacing d is nearly, but not completely, independent of rho. These results can be understood in terms of a simple charging model that reflects the competition between counterion entropy and inter-DNA (rho < 1) and interbilayer (rho > 1) repulsions. Finally, our approach and conclusions are compared with theoretical work by others, and with relevant experiments.
A comparison between a mean held theory of chain packing in membranes and micelles and Monte Carlo simulations is presented for model lipid bilayers. In both approaches the `'lipids'' are modeled as freely jointed (but self-avoiding) chains of spherical segments. The first segment of the chain represents the head group, anchored to the bilayer interface by a harmonic binding potential. The simulations are performed for symmetric bilayers composed of 200 chains, with periodic boundary conditions. Both pure and mixed bilayers (composed of long and short chains) are analyzed. In the simulation nonbonded segments interact via Lennard-Jones potentials, ensuring nearly uniform segment density in the bilayer core, as assumed in the mean held theory. The lateral pressure profiles governing the probability distribution of chain conformations in the mean field theory are related and compared to the tangential pressure profiles calculated from the simulations using Kirkwood-Buff's molecular theory. The two pressure profiles show very good agreement. We also calculate two conformational chain properties: end-segment distributions and orientational bond order parameters. The end-segment distributions calculated by the two approaches show excellent agreement. The order parameters compare somewhat less satisfactorily, yet we found that the order parameters derived from the simulations depend rather sensitively on the details of the interaction potential. In general, the results of the simulations support the use of the mean held theory as a (simple) tool for studying conformational chain statistics in confined environments and related thermodynamic properties, such as membrane curvature elasticity. (C) 1997 American Institute Physics.
A molecular level theory is presented for the thermodynamic stability of two (similar) types of structural complexes formed by (either single strand or supercoiled) DNA and cationic liposomes, both involving a monolayer-coated DNA as the central structural unit. In the `'spaghetti'' complex the central unit is surrounded by another, oppositely curved, monolayer, thus forming a bilayer mantle. The `'honeycomb'' complex is a bundle of hexagonally packed DNA-monolayer units. The formation free energy of these complexes, starting from a planar cationic/neutral lipid bilayer and bare DNA, is expressed as a sum of electrostatic, bending, mixing, and (for the honeycomb) chain frustration contributions. The electrostatic free energy is calculated using the Poisson-Boltzmann equation. The bending energy of the mixed lipid layers is treated in the quadratic curvature approximation with composition-dependent bending rigidity and spontaneous curvature. Ideal lipid mixing is assumed within each lipid monolayer. We found that the most stable monolayer-coated DNA units are formed when the charged/neutral lipid composition corresponds (nearly) to charge neutralization; the optimal monolayer radius corresponds to close DNA-monolayer contact. These conclusions are also valid for the honeycomb complex, as the chain frustration energy is found to be negligible. Typically, the stabilization energies for these structures are on the order of 1 k(B)T/Angstrom of DNA length, reflecting mainly the balance between the electrostatic and bending energies. The spaghetti complexes are less stable due to the additional bending energy of the external monolayer. A thermodynamic analysis is presented for calculating the equilibrium lipid compositions when the complexes coexist with excess bilayer.
Two structural-thermodynamic characteristics of cylindrical, wormlike micelles in dilute solution are studied using a molecular-level model: (a) the bending elasticity of the micelles and (b) their tendency to form intermicellar junctions (branches). The internal (free) energy of the micelles, before and after a bending deformation and junction formation, are calculated using mean field theories for the free energies of the molecules constituting these structures. The molecular free energies, which depend on the local packing geometries, include the contributions of head group repulsion forces, the hydrocarbon-water interfacial energy, and the chain conformational free energy. We find that when only the head group and surface contributions to the packing energy art:taken into account, the one-dimensional bending constant of the micelles is negligibly small, When the chain contribution is included, and when reasonable molecular packing parameters are used, we find that the persistence length, which is proportional to the bending rigidity, is typically a few tens of nanometers. The free energy change associated with the formation of a trijoint intermicellar junction upon the `'fusion'' of one micellar end cap with the cylindrical body of another micelle is found to be small but positive; about 10 k(B)T at room temperature. This conclusion does not refute ?he possibility that intermicellar junctions are metastable transients or that their formation may be favored entropically, due either to conformational degeneracy or excluded volume interactions between micelles. Our conclusions apply to aqueous solutions containing one, single-tail, surfactant species.
Activation energies for desorption and for diffusion were experimentally determined as a function of surface coverage for the system of ammonia on Re(001) utilizing optical second harmonic generation techniques, For the first time coverage grating with up to 5th order SH-diffraction is reported for K atoms on Re(001). Preliminary diffusion measurements were performed on this system as well. These systems may be considered as ideal model to study the effect of very strong lateral repulsion on the kinetics of desorption and diffusion. A MC study on the ammonia-Re(001) system is presented, which examines the significance of long range repulsive dipole-dipole interactions on the outcome desorption and diffusion kinetics, We found that a single set of parameters, within the dipole-dipole like (1/r(3)) dependence on adsorbates separation distance, explains qualitatively and in certain cases quantitatively the experimental observations. Interaction range up to 4th order neighbors must be computed in order to properly account for the results.
The effects of adsorbate lateral interactions on the kinetics of surface diffusion and desorption are studied by means of kinetic and thermodynamic Monte Carlo simulations. This study is motivated by recent diffusion and desorption experiments on the NH3/Re(001) system, which show that the activation energies of these processes decrease (in different fashions) with increasing surface coverage, the interactions between the adsorbates are thus assumed to be repulsive. A long range dipole-dipole-like potential is used to simulate both the diffusion and desorption processes. Most calculations are carried out with the interaction range extending up to fourth-order neighbors. Longer ranges are found to barely affect the kinetic behavior. On the other hand, shorter ranges of interaction result in qualitatively and quantitatively different structural (thermodynamic phase) behaviors and, consequently, in very different kinetics of diffusion and desorption. The model used to calculate diffusion kinetics assumes that the activation barrier to particle diffusion depends, simultaneously, on the local environments of both the initial and the final sites involved in the elementary event of particle jumps. The chemical diffusion coefficient is evaluated based on thermodynamic and kinetic Monte Carlo simulations. It is found to increase with surface coverage, reflecting the repulsive nature of the interactions. Yet, unlike the experimental results, the increase is nonmonotonic but rather, somewhat oscillatory-reflecting the structural phase transitions of the adsorbed layer. The activation energy of desorption is found to decrease by about 15 kcal/mole as the coverage increases from 0 to 1, showing steeper slopes around the coverages corresponding to a perfectly ordered adlayer phase. These results are in satisfactory qualitative and quantitative agreement with experiment. Finally, it is shown that the coverage dependence of the activation barrier to diffusion can be reasonably well evaluated from equilibrium thermodynamic desorption data. (C) 1996 American Institute of Physics.
A detailed treatment is provided of the various free-energy terms that contribute to the transfer of a polyalanine ct-helix from the aqueous phase into lipid bilayers. In agreement with previous work, the hydrophobic effect is found to provide the major driving force for helix insertion, However, an opposing effect of comparable magnitude is also identified and is attributed to the large free-energy penalty associated with the desolvation of peptide hydrogen bonds on transfer to the low dielectric environment of the bilayer. Lipid perturbation effects as well as the entropy loss associated with helix immobilization in the bilayer are also evaluated. Two configurations of a membrane-bound 25mer polyalanine helix were found to be lower in free energy than the isolated helix in the aqueous phase, The first corresponds to the case of vertical insertion, in which a helix terminus protrudes from each side of the bilayer. The second minimum is for the case of horizontal insertion, for which the helix is adsorbed upon the surface of the bilayer. The calculated free-energy minima are found to be in good agreement with recent measurements of related systems, Large free-energy barriers resulting from desolvation of unsatisfied hydrogen-bonding groups al the helix termini are obtained for both insertion processes. The barriers for insertion are significantly reduced if the helix termini are assumed to be `'capped'' through the formation of hydrogen bonds with polar sidechains, For uncapped helices, our results support recently proposed models in which helices are inserted by first adsorbing on the membrane surface and then having one terminus `'swing around'' so as to penetrate the bilayer,
We present an overview of the modern study of complex fluids which, because of the overwhelming breadth and richness of this field, unavoidably neglects many interesting systems and research developments. In proposing a definition of the field. we discuss first the special role played by phenomenological theory and the limitations of molecular-level description. The remainder of the article is organized into sections which treat model colloids, micellized surfactant solutions, interfacial films and microemulsions, bilayers and membranes, and new materials. In each instance we try to provide a physical basis for the special nature of interactions and long-range ordering transitions in these novel colloidal and thin layer systems. At the heart of understanding these highly varied phenomena lie the curvature dependence of surface energies and the coupling between self-assembly on small length scales and phase changes at large ones..
A statistical thermodynamic approach is used to analyze the various contributions to the free energy change associated with the insertion of proteins and protein fragments into lipid bilayers. The partition coefficient that determines the equilibrium distribution of proteins between the membrane and the solution is expressed as the ratio between the partition functions of the protein in the two phases. It is shown that when all of the relevant degrees of freedom (i.e., those that change their character upon insertion into the membrane) can be treated classically, the partition coefficient is fully determined by the ratio of the configurational integrals and thus does not involve any mass-dependent factors, a conclusion that is also valid for related processes such as protein adsorption on a membrane surface or substrate binding to proteins. The partition coefficient, and hence the transfer free energy, depend only on the potential energy of the protein in the membrane. Expressing this potential as a sum of a `'static'' term, corresponding to the equilibrium (minimal free energy) configuration of the protein in the membrane, and a `'dynamical'' term representing fluctuations around the equilibrium configuration, we show that the static term contains the `'solvation'' and `'lipid perturbation'' contributions to the transfer free energy. The dynamical term is responsible for the `'immobilization'' Free energy, reflecting the loss of translational and rotational entropy of the protein upon incorporation into the membrane. Based on a recent molecular theory of lipid-protein interactions, the lipid perturbation and immobilization contributions are then expressed in terms of the elastic deformation free energy resulting from the perturbation of the lipid environment by the foreign (protein) inclusion; The model is formulated for cylindrically shaped proteins, and numerical estimates are given for the insertion of an a-helical peptide into a lipid bilayer. The immobilization free energy is shown to be considerably smaller than in previous estimates of this quantity, and the origin of the difference is discussed in detail.